On the inverse problem for deformation rings of representations.

;; x ;; x Chapter 1 Introduction Let G be a finite group, let k be a finite field of characteristic p > 0. An n-dimensional representation of G over k is a group homomorphism : G! GL(k) n In the same way, if A is a complete Noetherian local ring with residue field k, an n-dimensional representation of G over A is a group homomorphism G! GL(A). We say that (A; ~ ) is a lift of ifA is a complete Noetherian n local ring with residue fieldk and ~ is a group homomorphism for which the diagram GL(A) n x x x x x x x x GL(k) G n commutes. Two lifts ~ ; ~ : G ! GL(A) of over A are said to be 12n equivalent if there exists a matrix K in ker(GL(A)! GL(k)) for which nn 1 K(g)K = ~ (g) for every g in G. An equivalence class of lifts is called 12 deformation of . Given a representation ofG overk and a complete Noethe- rian local ringA, we define the set Def(;A ) to be the set of all deformations of to A. For a representation : G! GL(k) the universal deformation ring R n a lift (R) for which the following universal property holds: for any lift u (A; ~ ) of there exists a unique homomorphism ’: R! A such that the following diagram: GL(R) n x x x ^ ’ x x x x x GL(A) G n u commutes, where the vertical arrow ^ ’ is the map induced by ’: R!A. //// ~ ~ ~ is ; u u